AI 中文总结
本研究证明特征零域上的无挠阿贝尔群可由其有理群域唯一确定，通过单项式缺陷群的自由性等结构完成重构，还得到了群域与群稳定化及里克德域的相关推论。
AI 中文摘要
对于一个无挠阿贝尔群$G$，令$K_{\boldsymbol{\text{Q}}}(G)=\text{Frac}\boldsymbol{\text{Q}}[G]$为其有理群代数的分式域。我们证明该域在同构意义下唯一确定群：$K_{\boldsymbol{\text{Q}}}(G)\boldsymbol{\text{同构于}}K_{\boldsymbol{\text{Q}}}(H)$当且仅当$G\boldsymbol{\text{同构于}}H$。主要的结构输入是：在每个特征零的域$k$上，单项式缺陷群$\boldsymbol{\text{Δ}}_k(G)=K_k(G)^\times/(k^\times X^G)$是自由阿贝尔群。我们给出一个自包含的证明：首先处理单变量普伊瑟克斯域$F(t^{\boldsymbol{\text{Q}}})$，从$F(t^{1/n!})$到$F(t^{1/(n+1)!})$的分解给出分裂包含，因为在特征零下代入的不可约元是无平方因子的；接着对可除包$\boldsymbol{\text{Q}}\boldsymbol{\text{⊗}}G$进行超限分解，从而证明一般情形。给定一个域同构，我们在公共乘法群内比较两个单项式子群，它们的交产生同构子群$M\boldsymbol{\text{≤}}G$和$N\boldsymbol{\text{≤}}H$，而商群$G/M$和$H/N$可嵌入自由缺陷群，因此是自由的；相对超越度表明这两个自由商群的秩相同，从而完成重构。作为推论，群域的有理函数稳定化恰好记录群的自由稳定化，且里克德的有界序列群产生一个域$F$满足$F\boldsymbol{\text{同构于}}F(x,y)$但$F\boldsymbol{\text{不}} \boldsymbol{\text{同构于}}F(x)$。
英文摘要
For a torsion-free abelian group $G$, let \[ K_{\mathbb{Q}}(G)=\operatorname{Frac} \mathbb{Q}[G] \] be the fraction field of its rational group algebra. We prove that this field determines the group up to isomorphism: \[ K_{\mathbb{Q}}(G) \cong K_{\mathbb{Q}}(H) \quad \Longleftrightarrow \quad G \cong H. \] The main structural input is that, over every field $k$ of characteristic zero, the monomial defect group \[ Δ_k(G)=K_k(G)^\times/(k^\times X^G) \] is free abelian. We give a self-contained proof. It first treats the one-variable Puiseux field $F(t^{\mathbb{Q}})$: factorization from $F(t^{1/n!})$ to $F(t^{1/(n+1)!})$ gives split inclusions because the substituted irreducibles are square-free in characteristic zero. A transfinite decomposition of the divisible hull $\mathbb{Q} \otimes G$ then proves the general case. Given a field isomorphism, we compare the two monomial subgroups inside the common multiplicative group. Their intersection produces isomorphic subgroups $M\leq G$ and $N\leq H$, while the quotients $G/M$ and $H/N$ embed in free defect groups and are therefore free. Relative transcendence degree shows that these two free quotients have the same rank, completing the reconstruction. As consequences, rational-function stabilization of group fields exactly records free stabilization of groups, and Rickard's bounded sequence group yields a field $F$ with $F\cong F(x,y)$ but $F\not\cong F(x)$.
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